PART F

PART F

2. The joint probability density function (pdf) of two random variables X and Y, Y f(x, y) = {a(1-x-y)+ 4xy, 0sxs1, 0sys1 0, elsewhere is non-negative in the unit square (shown) for any constant a, where Osas4 (a) Compute the marginal pdfs fx(x) and fy(y). Show all work! (5 pts) (b) For any fixed point (x,y) in the unit square, calculate the joint cumulative distribution function (cdf) F(x,y)=P(X SX, Y = y). Show all work! (5 pts) C) Use (b) to confirm that f(x,y) is indeed a legitimate joint pdf. Show all work! (3 pts) (d) Compute the marginal cdfs F/(x)=P(X 3x) and F-(y)=P(Y Sy). Show all work! (5 pts) (C) Formally calculate all the values of the constanta (Osas4), for which the variables X and Y are independent in the unit square. Show all work! No credit for random guessing! pts) (f) For each of the values in (e), confirm directly that the condition for independence is met, by explicitly checking the connection between... f(x,y) and its corresponding marginals fx (x) and f(y) (2 pts) F(x,y) and its corresponding marginals Fx(x) and F-6). (2 pts)